scholarly journals Implementation of the fractional-step, artificial compressibility with pressure projection (FSAC-PP) method into openfoam for unsteady flows

2021 ◽  
Vol 11 (5) ◽  
pp. 85-91
Author(s):  
Jesús Miguel Sánchez Gil ◽  
Tom-Robin Teschner ◽  
László Könözsy
Keyword(s):  
Time Derivative ◽  
Unsteady Flows ◽  
Time Discretization ◽  
Projection Methods ◽  
Fractional Step ◽  
Artificial Compressibility ◽  
Artificial Compressibility Method ◽  
Start Up ◽  
2D Flow ◽  
Pressure Projection

Commercial and open-source CFD solvers rely mostly on incompressible approximate projection methods to overcome the pressure-velocity decoupling, such as the SIMPLE (Patankar, 1980) or PISO (Issa, 1986) algorithm. Incompressible methods based on the Artificial Compressibility method (Chorin, 1967) lack a mechanism to evolve in time and need to be supplemented by a real time derivative through the dual time scheme. The current study investigates the implementation of the explicit dual time discretization of the Artificial Compressibility method into OpenFOAM and extends on that by applying the dual time scheme to the incompressible FSAC-PP method (Könözsy, 2012). Applied to the Couette 2D flow at Re=100 and Re=1000, results show that for both methods accurate time evolutions of the velocity profiles are presented, where the FSAC-PP methods seemingly produces smoother profiles compared to the AC method, especially during the start-up of the simulation.

Comparison of different time discretization schemes for solving the Allen–Cahn equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sana Ayub ◽  
Abdul Rauf ◽  
Hira Affan ◽  
Abdullah Shah
Keyword(s):  
Time Derivative ◽  
Time Discretization ◽  
Test Problems ◽  
Fractional Step ◽  
Cpu Time ◽  
Cahn Equation ◽  
Discretization Schemes ◽  
Conforming Finite Element ◽  
Derivative Term ◽  
Conforming Finite Element Method

Abstract This article aims to solve the nonlinear Allen–Cahn equation numerically. The diagonally implicit fractional-step θ-(DIFST) scheme is used for the discretization of the time derivative term while the space derivative is discretized by the conforming finite element method. The computational efficiency of the DIFST scheme in terms of CPU time and temporal error estimation is computed and compared with other time discretization schemes. Several test problems are presented to show the effectiveness of the DIFST scheme.

scholarly journals WEAK SOLUTIONS OF NAVIER–STOKES EQUATIONS CONSTRUCTED BY ARTIFICIAL COMPRESSIBILITY METHOD ARE SUITABLE

2011 ◽  
Vol 08 (01) ◽  
pp. 101-113 ◽  
Author(s):  
DONATELLA DONATELLI ◽  
STEFANO SPIRITO
Keyword(s):  
Fourier Transform ◽  
Weak Solutions ◽  
Stokes Equations ◽  
Time Derivative ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Artificial Compressibility ◽  
Artificial Compressibility Method

We prove that weak solutions constructed by artificial compressibility method are suitable in the sense of Scheffer. Using Hilbertian setting and Fourier transform with respect to time, we obtain non-trivial estimates on the pressure and the time derivative which allow us to pass to the limit.

scholarly journals A Unified Fractional-Step, Artificial Compressibility and Pressure-Projection Formulation for Solving the Incompressible Navier-Stokes Equations

2014 ◽  
Vol 16 (5) ◽  
pp. 1135-1180 ◽  
Author(s):  
László Könözsy ◽  
Dimitris Drikakis
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Benchmark Problems ◽  
Fractional Step ◽  
Navier Stokes Equations ◽  
Artificial Compressibility ◽  
Flow Problems ◽  
Dual Time Stepping ◽  
Pressure Projection ◽  
The Stability

AbstractThis paper introduces a unified concept and algorithm for the fractional-step (FS), artificial compressibility (AC) and pressure-projection (PP) methods for solving the incompressible Navier-Stokes equations. The proposed FSAC-PP approach falls into the group of pseudo-time splitting high-resolution methods incorporating the characteristics-based (CB) Godunov-type treatment of convective terms with PP methods. Due to the fact that the CB Godunov-type methods are applicable directly to the hyperbolic AC formulation and not to the elliptical FS-PP (split) methods, thus the straightforward coupling of CB Godunov-type schemes with PP methods is not possible. Therefore, the proposed FSAC-PP approach unifies the fully-explicit AC and semi-implicit FS-PP methods of Chorin including a PP step in the dual-time stepping procedure to a) overcome the numerical stiffness of the classical AC approach at (very) low and moderate Reynolds numbers, b) incorporate the accuracy and convergence properties of CB Godunov-type schemes with PP methods, and c) further improve the stability and efficiency of the AC method for steady and unsteady flow problems. The FSAC-PPmethod has also been coupled with a non-linear, full-multigrid and fullapproximation storage (FMG-FAS) technique to further increase the efficiency of the solution. For validating the proposed FSAC-PP method, computational examples are presented for benchmark problems. The overall results show that the unified FSAC-PP approach is an efficient algorithm for solving incompressible flow problems.

scholarly journals Goal-Oriented Error Estimation for the Fractional Step Theta Scheme

2014 ◽  
Vol 14 (2) ◽  
pp. 203-230 ◽  
Author(s):  
Dominik Meidner ◽  
Thomas Richter
Keyword(s):  
Time Discretization ◽  
Posteriori Error ◽  
Numerical Dissipation ◽  
Error Estimator ◽  
Fractional Step ◽  
Weighted Residual Method ◽  
Time Stepping ◽  
Backward Euler ◽  
Galerkin Formulation ◽  
Theta Method

Abstract. In this work, we derive a goal-oriented a posteriori error estimator for the error due to time-discretization of nonlinear parabolic partial differential equations by the fractional step theta method. This time-stepping scheme is assembled by three steps of the general theta method, that also unifies simple schemes like forward and backward Euler as well as the Crank–Nicolson method. Further, by combining three substeps of the theta time-stepping scheme, the fractional step theta time-stepping scheme is derived. It possesses highly desired stability and numerical dissipation properties and is second order accurate. The derived error estimator is based on a Petrov–Galerkin formulation that is up to a numerical quadrature error equivalent to the theta time-stepping scheme. The error estimator is assembled as one weighted residual term given by the dual weighted residual method and one additional residual estimating the Galerkin error between time-stepping scheme and Petrov–Galerkin formulation.

scholarly journals On the construction of suitable weak solutions to the 3D Navier–Stokes equations in a bounded domain by an artificial compressibility method

2017 ◽  
Vol 20 (01) ◽  
pp. 1650064 ◽  
Author(s):  
Luigi C. Berselli ◽  
Stefano Spirito
Keyword(s):  
Bounded Domain ◽  
Weak Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Bounded Domains ◽  
Suitable Weak Solutions ◽  
Navier Stokes Equations ◽  
Artificial Compressibility ◽  
Artificial Compressibility Method

We prove that suitable weak solutions of 3D Navier–Stokes equations in bounded domains can be constructed by a particular type of artificial compressibility approximation.

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